SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given equation for explicit formula
[tex]\begin{gathered} a_n=a_1\cdot r^{n-1} \\ where\text{ }a_1\text{ is the initial count\lparen first term\rparen} \\ r\text{ is the common ratio} \\ n\text{ is the number of years} \end{gathered}[/tex]
STEP 2: Write the given details
[tex]\begin{gathered} a_1=9000 \\ r=1+\frac{69}{100}=1.69\text{ since it is a growth rate} \\ \\ Hence,the\text{ equation is given as:} \\ a_x=9000(1.69)^{x-1} \end{gathered}[/tex]
STEP 3: Get the explicit equation for f(n)
n = x
Substitute n for x in the equation in step 2.
Therefore, the explicit equation is given as:
[tex]f(n)=9000\cdot(1.69)^{n-1}[/tex]
STEP 4: Answer part B
To get how many lionfish in the bay after 6 years
[tex]\begin{gathered} From\text{ equation above,} \\ n=6 \\ f(6)=9000\cdot(1.69)^{6-1} \\ f(6)=9000\cdot1.69^5 \\ f(6)=9000\cdot13.78584918 \\ f(6)=124072.6427 \\ f(6)\approx124073 \end{gathered}[/tex]
Hence, there will be approximately 124073 lionfish
STEP 5: Get the recursive formula
1400 lionfish was removed per year, this gives an equation defined below:
Recursive formula is given as
[tex]a_n=r(a_{n-1})[/tex]
Since we know that the difference each year is 1400, this gives the equation below:
[tex]a_n-1400[/tex]
By substitution, the recursive formula will be given by:
Since 1400 is removed each year, we have:
[tex]\begin{gathered} f(n)=a_{n-1}-1400n \\ f(n)=9000\cdot(1.69)^{n-1}-1400n \end{gathered}[/tex]
